People often point at how many cells make mistakes and say just what they said in the video "Boy its a miracle we dont all get cancer all the time!".
An overlooked, but very important factor, is that we also have very strict systems that keep a cell from proliferating as a cancer. A cell that develops a defect in the genes that is not corrected, but still has working oncosuppressor genes will simply go into apoptosis.
Cancer needs specific mutations that disable these failsafe mechanics.
But that is only part of the road that leads to cancer. The proto-cancer cell needs to also evade the immune system, which is trained to find and destroy cells that are not doing what they are suppose to do.
These mutations cannot just be haphazard as highly unstable cancers with major errors in proof-reading can have a better prognosis because their radical protein productions suddenly draws the attention of the immune system.
Besides that it will also need pro-oncogens, to give it the proliferation advantage over neighbouring tissue. Not to mention all other kinds of mutations that stretch the telomers, allow it to invade tissue, and spread through the body.
A cancer is not the result of any random mistake, it is a very specific series of errors.